Question

5) Find the points on the Plane curve given by rcE)= <cos"e), since)> at which the curue is not Sm o0th 6) Pind the orc 1ergth of the curue 33 t 2 8Find the coruotore of the curve given 2 4 Find Corve res 3 Cose Tt 3sin cEjt 46 R unit Norma vectof of the the 10 Pind the Tongent 3 NOrma co Mponent Of the accelerationof the curve oiven by 2

Answer 1

Find be points on the plane curve Given by r (t) = (cos^4 (t), sin^4 (t)) at which the curve is not smooth Given that r (t) = (cos^2 (t), sin^4 (t)) i.e x (t) = cos^4 t, y (t) = sin^4 t. The curve is not smooth at its singular points. The condition for finding singular points is x' (t) = y' (t) = 0 Now x' (t) = -4 cos^3 t sin t y' (t) = 4 sin^3 t cos t clearly, We show that x' (t) & y' (t) are zero at t = nx, (2n + 1) x/2 Because sin nx = 0 Forall n, also cos (2n + 1) pi/2 = 0 Forall n Hence the Given curve is not smooth at t = npi (2n + 1) pi/2 n elementof N (6) Find the are length of the curve r (t) = 1/3 t^3 i + squareroot 2/2 t^3 j + tk 0 lessthanorequalto t lessthanorequalto 2.